Place probabilities in normal order statistics models for horse races

Abstract

Independent observations X 0, X 1…, X N+1 are drawn from each of N populations whose distribution functions F(x – θi ) have means θ i , 0 ≦ i < N, and we wish to calculate the probability P k;N that X 0 is the k th largest observation. For normal populations an approximation is given for P K;N based on a Taylor series expansion in the θ 's. If F(x) has an increasing failure rate, as is the case for the normal, an upper bound can be given for the ‘win' probability P 1;N Some moment relations for normal order statistics are also given.